First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt:
$$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$
Secondly, is there a (possibly infinite) set of sentences $T$ which has the infinite structures as models? I think it has something to do with Tarskis definition of truth. I use the following notation: $M\vDash \phi$ means $M$ satifies $\phi$, i.e the sentence $\phi$ is valid in $M$
Thirdly, how would you argue that there is no sentence $\phi$ which has the finite structures as models, I mean without a concret proof?